Existence of finite dimensional ample subspace

Question

This result is from M F Atiyah , K Theory. Let $E$ be a vector bundle over $X, ~ \Gamma(E)$ be set of sections of $E$. An ample subspace is a subspace $V$ of $\Gamma(E)$ such that the map $\varphi: X \times V \to E$ defined as $\varphi(x,s)=s(x)$ is surjective. For trivial bundle there exist a finite dimensional ample subspace.

Atiyah gives the following result : If $E$ is any bundle over a compact Hausdorff space then $\Gamma(E)$ contains a finite dimensional ample subspace.

I have some doubts in the proof given below :

By compactness of $X$ we can take finite open cover $\{U_{\alpha } \}$ such that $E|U_{\alpha}$ is trivial. Then there exist finite dimensional ample subspace $V_{\alpha}$ of $\Gamma(E|U_{\alpha})$. Also there exists partitions of unity $\{ p_{\alpha} \}$ dominated by $U_{\alpha}$.

Define $\theta_{\alpha}: V_{\alpha} \to \Gamma(E)$ such that $ \theta_{\alpha}(s)(x)= p_{\alpha}(x) s_{\alpha}(x)$ when $x \in U_{\alpha}$ and $0$ otherwise.

Then the book says the $\theta_{\alpha}$ defines a homomorphism

$\Theta: \prod V_{\alpha} \to \Gamma(E)$. Image of $\Theta$ is the required ample subspace.

My doubts are mainly the following

1. Does partitions of unity has some role in the definition of $\theta_{\alpha}$ ? Usually partition of unity is used to prove that the function agrees on the intersection. But here we don't have intersection for $U_{\alpha}$ and its compliment.
2. How does $\Theta$ is defined ? I was thinking $\Theta((s_{\alpha}))(x)= \sum \theta_{\alpha}(s)(x)$ Is it correct? Then on points $x \in X$ such that $x \in U_{\alpha}$ for exactly one $\alpha$ the value of $\Theta((s_{\alpha}))(x)= p_{\alpha}(x)s_{\alpha}(x)$. Mean while if $x \in U_{\alpha} \cap U_{\beta}$ then $\Theta((s_{\alpha}))(x)= p_{\alpha}(x)s_{\alpha}(x)+p_{\beta}(x)s_{\beta}(x)$.
3. Can we construct $U_{\alpha}$ such that they are disjoint ?

Answer 1

1. To define $\theta_\alpha$ we can take any $s_\alpha : X \to [0,1]$ whose support is contained in $U_\alpha$. Then for each $s \in V_\alpha$ the function $\theta_\alpha(s) : X \to E$ is a continuous section. However, that the $s_\alpha$ form a partition of unity subordinated to $\{U_\alpha\}$ will be needed later.

2. Your definition of $\Theta$ is correct. Note that the sum is finite. But $\Theta((s_{\alpha}))(x)= p_{\alpha}(x)s_{\alpha}(x)+p_{\beta}(x)s_{\beta}(x)$ is satisfied only if $x \notin U_{\gamma}$ for all $\gamma \ne \alpha,\beta$.

3. You cannot expect that the $U_\alpha$ are pairwise disjoint. If you have more than one $U_\alpha$, this would imply that $X$ is not connected. However, there exist compact connected $X$ with non-trivial vector bundles.